Lecture 1, on Monday January 26,
was devoted to trefoils. A trefoil was identified with the
curve defined in the unit sphere S3 of the complex plane
C2=R4 by equation
z3=w2. It was presented as a generic fiber of
a Seifert fibration S3->S2=CP1:
(z,w)|->(z3:w2).

The partition of the 3-sphere to fibers of this map was considered.
We cosiderd also a similar Hopf fibration S3->S2=CP1:
(z,w)|->(z:w).

Then we discussed geometry of the space of complex cubic polynomials in
one variable. After factoring out a punctured plane (by dividing by the
coefficient at the cubic term) and a plane (by translating of the plane
of roots killing the term of degree 2), we identified the space of
polynomials X3+pX+q with C2 and the
subspace of polynomials with multiple roots with the complex curve defined
by 4p3+27q2.

Then the space of unordered triples of
points in the plane C considered up to translations and dilations was identified with
the 3-sphere plus a point. (The point corresponds to the triple in which
all 3 points coincide.) The triples in which 2 of 3 points coincide and the
third one is different correspond under this identification to a trefoil.

The fundamental group of unordered triples of pairwise distinct points
is the 3 string braid group B3.
Therefore the group of 3 string braids is identified with the group of a
trefoil.

Lecture 2, on Wednesday January 28, started with revisiting
and generalizations of theorems presented in the first lecture.
Trefoil was partly replaced by torus knots and links.

Artin braid groups: geometric braids, their equivalence, multiplication.
Braid groups as the fundamental groups and as the group of isotopy classes
of homeomorphisms of disk fixed on the boundary and preserving a finite set
of interior points.

The group of a knot. The groups of torus links.

Lecture 3, Monday February 2.
Stratification of the space of unordered n-tuples of pairwise distinct
points and the standard presentation of the braid group.
The isotopy classes of homeomorphisms corresponding to the standard generators of the braid group.
Dehn twists.

Homomorphism of a braid group onto symmetric group. Pure braids.
Their interpretations via configuration spaces of ordered collections of
points and isotopy classes of homeomorphisms.
Short exact sequence

0 -> Pn -> Bn -> Sn -> 0.

Homomorphisms Pn -> Pn-1 and Pn-1
-> Pn. Short exact sequence

0 -> Fn-1 -> Pn -> Pn-1 ->
0.

Centers of the braid groups. Coverings related to the sequences. Higher
homotopy groups of the configuration spaces. Eilenberg-MacLane spaces.
Homology of the braid groups.

Lecture 4, on Wednesday February 4,
was devoted to the little ones of Topology.
Topological Classification of 1-manifolds.
Mapping class groups of the line and circle. The fundamental group
of Homeo(S1).

Topology as the only field in Mathematics which hesitates its
own finite objects.

How many points are needed for a non-trivality of the fundamental
group?
Digital circle and digital line. The fundamental group of the digital
circle.

Digital plane and digital Jordan Theorem.

Lecture 5, Monday February 9.

Lost Chater of General Topology.
Topological structure can be interpreted as the set of all continuous
maps from the space to the connected pair of points.

Axioms of topological structure then mean that the set of maps
is closed with respect to operations of taking supremum, minimum
and contains constant maps.

The space of simplices of a simplicial space. Recovering of the
simplicial space out of its space of simplices. Weak homotopy equivalence
between a simplicial space and its space of simplices. The space of cells of
a polyhedron.

T0-equivalence of points in a topological space.
Preordering of points. T0-spaces as posets.
Structure of a general finite topological space.

Theorem. Any finite topological space is weak homotopy equivalent
to a compact polyhedron.

Lecture 6, Wednesday February 11.

Baricentric subdivision of a triangulated space. Baricentric subdivision
of a poset. Baricentric subdivision turns any T0 finite
topological space to the space of simplices of a compact simplicial space.

Projective duality. Dual plane curves. Duality between an inflection
point and a cusp. Surface dual to a spatial curve.

Moment maps of a toric variety to the corresponding convex polyhedron.

Lecture 9, Monday February 23.

The Newton polyhedron of a polynomial. The hypersurface of a toric
variety defined by a polynomial. Singular points of a hypersurface. Is a
generic hypersurface singular? If one considers all polynomials of fixed
degree, then yes. If the Newton polyhedron is fixed, then in affine or
projective space the answer may be no. In what sense a generic toric variety
is non-singular in the toric variety defined by its Newton polyhedron.